The term "airport" refers to the entire aerodrome, including the runways and related buildings. However, when you say "in" an airport, that tends to refer to the buildings, because the runways etc are outside. So, saying a plane landed "in" an airport is incorrect, it should be "we've just touched down at the airport".
When it comes to the name of a town, city or country, we say that we are "in" it, so it would be correct to say "we've just touched down in Qatar".
The only context where "in" and "at" becomes a choice is if you are distinguishing between being within the boundaries, or at the perimeter of something. For example, you might arrive at a building by car, and then go in on foot.